Analytical Solutions for Brownian Coagulation of Fractal Aggregates Based on Power-Law Relaxation Approximation

Existing analytical solutions for Brownian coagulation of fractal aggregates typically assume a constant geometric standard deviation during derivation to achieve closed-form expressions. This simplification introduces notable systematic deviations and limits their applicable ranges. The present study develops a power-law relaxation approximation to address this issue and derives corresponding analytical solutions for both the continuum and free-molecular regimes using the log-normal method of moments. The proposed analytical solutions reduce to the existing analytical expressions as the relaxation coefficient approaches zero and converge to the asymptotic solutions as the relaxation coefficient approaches infinity. This demonstrates that the two conventional models are unified within a single theoretical framework. The relaxation solutions are validated against numerical reference results across a wide range of mass fractal dimensions and initial geometric standard deviations. The average relative errors remain below 2.1% for all test cases, confirming that the proposed formulations achieve substantially higher accuracy than existing analytical solutions. The mass fractal dimension exerts only a slight effect on Brownian coagulation in the continuum regime, whereas it acts as a dominant factor in the free-molecular regime. For the latter regime, smaller fractal dimensions lead to larger collision cross-sections and substantially accelerate the size growth of aggregates.